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Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution of Prime Numbers

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Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution of Prime Numbers COALESCENCE PHENOMENON OF QUANTUM COHOMOLOGY OF GRASSMANNIANS AND THE DISTRIBUTION OF PRIME NUMBERS GIORDANO COTTI(†) Abstract. The occurrence and frequency of a phenomenon of resonance (namely the coalescence of some Dubrovin canonical coordinates) in the locus of Small Quantum Cohomology of complex Grass- mannians is studied. It is shown that surprisingly this frequency is strictly subordinate and highly influenced by the distribution of prime numbers. Two equivalent formulations of the Riemann Hypoth- esis are given in terms of numbers of complex Grassmannians without coalescence: the former as a constraint on the disposition of singularities of the analytic continuation of the Dirichlet series associ- ated to the sequence counting non-coalescing Grassmannians, the latter as asymptotic estimate (whose error term cannot be improved) for their distribution function. Contents Notations 2 1. Introduction and Results3 1.1. Plan of the paper. 6 2. Frobenius Manifolds and Quantum Cohomology6 2.1. Semisimple Frobenius Manifolds8 2.2. Gromov-Witten Theory and Quantum Cohomology9 3. Quantum Satake Principle 11 3.1. Results on classical cohomology of Grassmannians 11 3.2. Quantum Cohomology of G(k, n) 15 4. Frequency of Coalescence Phenomenon in QH•(G(k, n)) 16 4.1. Results on vanishing sums of roots of unity 17 4.2. Characterization of coalescing Grassmannians 18 4.3. Dirichlet series associated to non-coalescing Grassmannians, and their rareness 19 5. Distribution functions of non-coalescing Grassmannians, and equivalent form of the Riemann Hypothesis 22 References 24 arXiv:1608.06868v2 [math-ph] 21 Sep 2016 (†) SISSA, Via Bonomea, 265 - 34136 Trieste ITALY E-mail address: [email protected]. 1 • 2 COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS Notations Given two natural numbers 0 < k < n we will denote by G(k, n) the Grassmannian of k-dimensional -subspaces of n. Thus (1, n) = n−1. C C G PC In what follows we will use the following notations for number theoretical functions: • P1(n) := min {p ∈ N: p is prime and p|n} , n ≥ 2; • for real positive x, y we define Φ(x, y) := card ({n ≤ x: n ≥ 2,P1(n) > y}); P α • πα(n) := p prime p , α ≥ 0; p≤n • ζ(s) is the Riemann ζ-function; • ζ(s, k) will denote the truncated Euler product −1 Y 1 ζ(s, k) := 1 − , k ∈ , s ∈ \{0} . ps R>0 C p prime p≤k • ζP (s) is the Riemann prime ζ-function, defined on the half-plane Re(s) > 1 by the series X 1 ζ (s) := ; P ps p prime • ζP,k(s) will denote the partial sums X 1 ζ (s) := ; P,k ps p prime p≤k • ω : R≥1 → R is the Buchstab function ([Buc37]), i.e. the unique continuous solution of the delay differential equation d (uω(u)) = ω(u − 1), u ≥ 2, du with the initial condition 1 ω(u) = , for 1 ≤ u ≤ 2. u If f, g : R+ → R, with g definitely strictly positive, we will write • f(x) = Ω+(g(x)) to denote f(x) lim sup > 0; x→∞ g(x) • f(x) = Ω−(g(x)) to denote f(x) lim inf < 0; x→∞ g(x) • f(x) = Ω±(g(x)) if both f(x) = Ω+(g(x)) and f(x) = Ω−(g(x)) hold. • COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS 3 1. Introduction and Results In this paper we exhibit a direct connection between the theory of Frobenius Manifolds, more precisely the Gromov-Witten and Quantum Cohomology theories, and one of the most ancient problems lying at the heart of Mathematics, the distribution of prime numbers. We study a phenomenon of resonance in the small quantum cohomology of complex Grassmannians (consisting in the coalescence of some natural parameters there defined), and show that the occurrence and frequency of this phenomenon is surpris- ingly related to the distribution of prime numbers. This relation is so strict that it leads to (at least) two equivalent formulations of the famous Riemann Hypothesis: the former is given as a constraint on the disposition of the singularities of a generating function of the numbers of Grassmannians not presenting the resonance, the latter as an (essentially optimal) asymptotic estimate for a distribution function of the same kind of Grassmannians. Besides their geometrical-enumerative meaning, three point genus zero Gromov-Witten invariants of complex Grassmannians implicitly contain information about the distribu- tion of prime numbers. This mysterious relation deserves further investigations. Born in the last decades of the XX-th century, in the middle of the creative impetus for a mathemat- ically rigorous foundations of Mirror Symmetry, the theory of Frobenius Manifolds ([Dub96], [Dub98], [Dub99], [Man99], [Her02], [Sab08]) seems to be characterized by a sort of universality (see [Dub04]): this theory, in some sense, is able to unify in a unique, rich, geometrical and analytical description many aspects and features shared by the theory of Integrable Systems, Singularity Theory, Gromov-Witten In- variants, the theory of Isomondromic Deformations and Riemann-Hilbert Problems, as well as the theory of special functions like Painlevé Transcendents. Originally introduced by physicists ([Vaf91]), in the context of N = 2 Supersymmetric Field Theo- ries and mirror phenomena, the Quantum Cohomology of a complex projective variety X (or more in general a symplectic manifold [MS12]) is a family of deformations of its classical cohomological algebra • L k • structure defined on H (X) := k H (X; C), and parametrized over an open domain D ⊆ H (X): the ∼ • fiber over p ∈ D is identified with the tangent space TpD = H (X). This is exactly the prototype of a Frobenius manifolds, namely a manifold, endowed with a flat metric, on whose tangent spaces there is a well defined commutative, associative, unitary algebra structure, satisfying also some more compatibility conditions. The structure constants of the quantum deformed algebras are given by (third derivatives of) X a generating function F0 of Gromov-Witten Invariants of genus 0 of X: these rational numbers morally “count” (modulo parametrizations) algebraic/pseudo-holomorphic curves of genus 0 on X, with a fixed degree, and intersecting some fixed subvarieties of X. Focusing on the case of complex Grassmannians, X = G(k, n), in what follows we study the occurrence of a phenomenon of coalescence of some numerical parameters on their small quantum cohomology (i.e. D ∩ H2(X)). This set of parameters defines a system of coordinates (called Dubrovin canonical coordi- nates) near any point whose corresponding Frobenius algebra is semisimple. It is important to keep in mind that these parameters have a double algebro-geometric meaning: (1) at each point p ∈ D their vector fields coincide with the idempotents of the algebra structure defined on TpD; (2) they are the eigenvalues1 of an operator of multiplication by a distinguished vector field (the Euler vector field) which codifies the grading structure of the algebras. Notice that, along the small quantum locus, the Euler vector field coincides with the first Chern class c1(X). Moreover, this coalescence of canonical coordinates can be interpreted as a “delicate point”2 in the local description of semisimple Frobenius manifolds as spaces of deformation parameters for isomonodromic families of linear differential systems with rational coefficients in complex domains, at least as exposed 1The canonical coordinates are not uniquely determined by the requirement (1) alone: there is a shift ambiguity. We fix this freedom by requirement (2). 2In the theory of Painlevé equations they are called critical points. • 4 COALESCENCE OF QH (G(k, n)) AND THE DISTRIBUTION OF PRIME NUMBERS in [Dub98] and [Dub99]. Such a description is carried on in all details in [CG16], for the general analytic case, and in [CDG16] for the specific Frobenius case. For simplicity, we will call coalescing a Grassman- nian such that some of the Dubrovin canonical coordinates coalesce. The questions, to which we answer in the present paper, are the following: (i) For which k, n the Grassmannian G(k, n) is coalescing? (ii) How frequent is the phenomenon of coalescence among all Grassmannians? Let us summarize some of the main results obtained. Theorem (cf. Theorems 4.4, 4.5, 5.1, 5.2 and Corollaries 4.3, 4.2 for more details) PART I The complex Grassmannian G(k, n) is coalescing if and only if P1(n) ≤ k ≤ n − P1(n). In particular, all Grassmannians of proper subspaces of Cp, with p prime, are not coalescing. PART IILet us denote by l˜n, for n ≥ 2, the number of non-coalescing Grassmannians of proper subspaces of Cn, i.e. l˜n := card {k : G(k, n) is not coalescing} , and let ∞ X l˜n Le (s) := ns n=2 be the associated Dirichlet series. Le (s) is absolutely convergent in the half-plane Re(s) > 2, where it can be represented by the infinite series involving the Riemann zeta function and the truncated Euler products X p − 1 2ζ(s) Le (s) = − 1 . ps ζ(s, p − 1) p prime By analytic continuation, Le (s) can be extended to (the universal cover of) the punctured half-plane ( ) ρ ρ pole or zero of ζ(s), {s ∈ : Re(s) > σ}\ s = + 1: , C k k squarefree positive integer 1 X 3 σ := lim sup · log l˜k , 1 ≤ σ ≤ , n→∞ log n 2 k≤n k composite having logarithmic singularities at the punctures. In particular, we have the equivalence of the following statements: 1 • (RH) all non-trivial zeros of the Riemann zeta function ζ(s) satisfy Re(s) = 2 ; 0 • the derivative Le (s) extends, by analytic continuation, to a meromorphic function in the half- 3 plane 2 < Re(s) with a single pole of oder one at s = 2.
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